An explicit classification of dual pairs in exceptional Lie algebras

Abstract

The primary goal of this paper is to explicitly write down all semisimple dual pairs in the exceptional Lie algebras. (A dual pair in a reductive Lie algebra $\mathfrak{g}$ is a pair of subalgebras such that each member equals the other's centralizer in $\mathfrak{g}$.) In a 1994 paper, H. Rubenthaler outlined a process for generating a complete list of candidate dual pairs in each of the exceptional Lie algebras. However, the process of checking whether each of these candidate dual pairs is in fact a dual pair is not easy, and requires several distinct insights and methods. In this paper, we carry out this process and explain the relevant concepts as we go. We also give plenty of examples with the hopes of making Rubenthaler's 1994 result not only more complete but more usable and understandable.